Chapter 1 Section 4

Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Real Numbers and the Number Line Classify numbers and graph them on number lines. Tell which of two real numbers is less than the other. Find the additive inverse of a real number. Find the absolute value of a real number. Interpret the meanings of real numbers from a table of data

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Classify numbers and graph them on number lines. Objective 1 Slide 1.4-3

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Classify numbers and graph them on a number line. Natural numbers (or counting numbers) and whole numbers, along with many others, can be represented on a number line like the one below. We draw a number line by choosing any point on the line and labeling it 0. Then we choose any point to the right of 0 and label it 1. The distance between 0 and 1 gives a unit of measure used to locate, and then label other points. The “arrowhead” is used to indicate the positive direction on a number line. Slide 1.4-4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. The natural numbers are located to the right of 0 on the number line. For each natural number, we can place a corresponding number to the left of 0. Each is the opposite, or negative, of a natural number. Positive numbers and negative numbers are called signed numbers. The natural numbers, their opposites, and 0 form a new set of numbers called the integers. {..., −3, −2, −1, 0, 1, 2, 3,... } Slide Classify numbers and graph them on a number line. (cont’d) The three dots ( … ) show that the list of numbers continues in the same way indefinitely.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use an integer to express the number in boldface italics in each application. Erin discovers that she has spent $53 more than she has in her checking account. The record-high Fahrenheit temperature in the United States was 134° in Death Valley, California, on July 10, (Source: World Almanac and Book of Facts.) Solution: −53 Solution: 134 Slide EXAMPLE 1 Using Negative Numbers in Applications

Copyright © 2012, 2008, 2004 Pearson Education, Inc. { is a quotient of two integers, with denominator not 0} is the set of rational numbers. (Read as “the set of all numbers x such that x is a quotient of two integers, with denominator not 0.”) Since any number that can be written as the quotient of two integers is a rational number, all integers, mixed numbers, terminating (or ending decimals), and repeating decimals are rational. This is called set-builder notation. This notation is convenient to use when it is not possible to list all the elements of a set. Slide Classify numbers and graph them on a number line. (cont’d)

Copyright © 2012, 2008, 2004 Pearson Education, Inc. To graph a number, we place a dot on the number line at the point that corresponds to the number. The number is called the coordinate of the point. { is a nonrational number represented by a point on the number line} is the set of irrational numbers. The decimal form of an irrational number neither terminates nor repeats. { is a rational or an irrational number} is the set of real numbers. Slide Classify numbers and graph them on a number line. (cont’d)

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Identify each real number in the set as rational or irrational. Solution: are rational; and −π are irrational Slide EXAMPLE 2 Determining Whether a Number Belongs to a Set

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Tell which of two real numbers is less than the other. Objective 2 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Ordering of Real Numbers For any two real numbers a and b, a is less than b if a is to the left of b on the number line. This means that any negative number is less than 0, and any negative number is less than any positive number. Also, 0 is less than any positive number. We can also say that, for any two real numbers a and b, a is greater than b, if a is to the right of b on the number line. Slide Tell which of two real numbers is less than the other.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: False Determine whether the statement is true or false. Slide EXAMPLE 3 Determining the Order of Real Numbers

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the additive inverse of a real number. Objective 3 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. By a property of the real numbers, for any real number x (except 0), there is exactly one number on the number line the same distance from 0 as x, but on the opposite side of 0. Such pairs are called additive inverses, or opposites, of each other. Additive Inverse The additive inverse of a number x is the number that is the same distance from 0 on the number line as x, but on the opposite side of 0. Double Negative Rule For any real number x, −(−x) = x. Slide Find the additive inverse of a real number. The additive inverse of −7 is written −(−7) and can be read “the opposite of −7” or “the negative of −7” or 7.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the absolute value of a real number. Objective 4 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Distance is a physical measurement, which is never negative. Therefore, the absolute value of a number is never negative. The absolute value of a real number can be defined as the distance between 0 and the number on the number line. The symbol for the absolute value of the number x is |x|, read “the absolute value of x.” Absolute Value For any real number x,. The “−x ” in the second part of the definition does NOT represent a negative number. Since x is negative in the second part, −x represents the opposite of a negative number—that is, a positive number. The absolute value of a number is never negative. Slide Find the absolute value of a real number.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: 32 Simplify by finding the absolute value. Slide EXAMPLE 4 Finding the Absolute Value 32 −32 −30

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Interpret the meanings of real numbers from a table of data. Objective 5 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: gasoline, from 2005 to 2006 In the table, which category in which year represents the greatest percent increase? Slide EXAMPLE 5 Interpreting Data